Sphere hydrodynamic interactions between

Marlow and Rowell discuss the deviation from Eq. V-47 when electrostatic and hydrodynamic interactions between the particles must be considered [78]. In a suspension of glass spheres, beyond a volume fraction of 0.018, these interparticle forces cause nonlinearities in Eq. V-47, diminishing the induced potential E. 

At finite concentration, tire settling rate is influenced by hydrodynamic interactions between tire particles. For purely repulsive particle interactions, settling is hindered. Attractive interactions encourage particles to settle as a group, which increases tire settling rate. For hard spheres, tire first-order correction to tire Stokes settling rate is given by [33]... 

Hindered Settling When particle concentration increases, particle settling velocities decrease oecause of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered setthng is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore Br. J. Appl. Fhys., 9, 477—482 [1958]) give, for uniformly sized spheres,... 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

Mazur (1982) and Mazur and van Saarloos (1982) developed the so-called method of induced forces in order to examine hydrodynamic interactions among many spheres. These forces are expanded in irreducible induced-force multipoles and in a hierarchy of equations obtained for these multipoles when the boundary conditions on each sphere were employed. Mobilities are subsequently derived as a power series-expansion in p 1. In principle, calculations may be performed to any order, having been carried out by the above authors through terms of 0(p 7) for a suspension in a quiescent fluid. To that order, hydrodynamic interactions between two, three, and four spheres all contribute to the final result. This work is reviewed by Mazur (1987). 

The long-range, purely hydrodynamic interaction between two suspended spheres in a shear flow was first calculated by Guth and Simha (1936), yielding a value of kx = 14.1 via a reflection method. Saito (1950,1952) proposed two alternative modifications, obtaining kt = 12.6 and 2.5, respectively the latter value is obtained upon supposing a spatially uniform distribution of particles. 

For concentrated suspensions, hydrodynamic interactions among particles must be considered. The hydrodynamic interactions between spherical particles can be taken into account by means of a cell model, which assumes that each sphere of radius a is surrounded by a virtual shell of outer radius b and the particle volume... 

Here, the superscript zero signifies that U and are the velocities that the spheres would have in the absence of any hydrodynamic interaction between them. Because the particles are identical,... 

Relative Viscosity of Suspensions One of the most interesting derivations of the T vs. (() dependence (covering the full range of concentration) was published by Simha [1952]. He considered the effects of concentration on the hydrodynamic interactions between suspended particles of finite size. (Note that previously the particles were simply considered point centers of force that decayed with cube of the distance.) Simha adopted a cage model, placing each solid, spherical particle of radius a inside a spherical enclosure of radius b. At distances x < b, the presence of other particles does not influence flow around the central sphere and the Stokes relation is satisfied. This assumption leads to a modified Einstein [1906, 1911] relation ... 

In reality, at any finite concentration, flows around near-neighbor spheres are likely to interfere with each other, so to compare Eqs. (1.18) and (1.19) with experiment, we need to extrapolate to zero concentration to eliminate the contribution of such hydrodynamic interactions between particles ... 

When filler spheres are close enough, hydrodynamic interactions between particles need to be considered. By factoring in the effects of two-body interactions, a modified equation was derived that is apphcable up to > = 0.1 (24). 

Consider a molecular liquid with Newtonian behaviour (see Chapter 4) such as water, benzene, alcohol, decane, etc. The addition of a spherical particle to the liquid will increase its viscosity due to the additional energy dissipation related to the hydrodynamic interaction between the liquid and the sphere. Further addition of spherical particles increases the viscosity of the suspension linearly. Einstein developed the relationship between the viscosity of a dilute suspension and the volume fraction of solid spherical particles as follows (Einstein, 1906) ... 

Einstein s analysis was based on the assumption that the particles are far enough apart so that they do not influence each other. Once the volume fraction of solids reaches about 10%, the average separation distance between particles is about equal to their diameter. This is when the hydrodynamic disturbance of the liquid by one sphere begins to influence other spheres. In this semi-dilute concentration regime (about 7-15 vol% solids), the hydrodynamic interactions between spheres results in positive deviation for Einstein s relationship. Batchelor (1977) extended the analysis to include higher order terms in volume fraction and found that the suspension viscosities are still Newtonian but increase with volume fraction according to ... 

The most appropriate particle size to use in equations relating to fluid-particle interactions is a hydrodynamic diameter, i.e. an equivalent sphere diameter derived from a measurement technique involving hydrodynamic interaction between the particle and fluid. In practice, however, in most industrial applications sizing is done using sieving and correlations use either sieve diameter, Xp or volume diameter, Xy, For spherical or near spherical particles Xv is equal to Xp. For angular particles, Xy l.lSXp. 

Drainage experiments conducted using the AFM and SEA, which have the ability to probe fluid films of nano- and molecular thicknesses, may be able to yield detailed information on the superhydrophobic slip. A lot of data is already available. " The recent theory of hydrodynamic interaction between disks shed some light on what could happen qualitatively, but cannot be used for a quantitative analysis of the hydrodynamic data obtained with the AFM and SEA. The same remark concerns the use of a theory of a film drainage between smooth hydrophobic surfaces, which is not fully applicable to quantify a superhydrophobic slip. We believe that a challenge for a theory would be to extend a theoretical modeling of experimentally relevant sphere versus plane geometry. This will open many possibilities for new experiments and could revolutionize the field. 

The higher order dependence on for spheres, in eq. 10.2.15, is shown in Figure 10.2.3. This departure from the Einstein equation is due to hydrodynamic interactions between spheres and to other interparticle forces. We will examine these effects in Section 10.4, but first we look at the influence of particle shape on the rheology of dilute suspensions. 

Modern synthetic methods allow preparation of highly monodisperse spherical particles that at least approach closely the behavior of hard-spheres, in that interactions other than volume exclusion have only small influences on the thermodynamic properties of the system. These particles provide simple model systems for comparison with theories of colloidal dynamics. Because the hard-sphere potential energy is 0 or 00, the thermodynamic and static structural properties of a hard-sphere system are determined by the volume fraction of the spheres but are not affected by the temperature. Solutions of hard spheres are not simple hard-sphere systems. At very small separations, the molecular granularity of the solvent modifies the direct and hydrodynamic interactions between suspended particles. 

In order to clarify the detailed character of the hydrodynamic interactions between colloids in SRD, Lee and Kapral [103] numerically evaluated the fixed-particle friction tensor for two nano-spheres embedded in an SRD solvent They found that for intercolloidal spacings less than 1.2 d, where d is the colloid diameter, the measured friction coefficients start to deviate from the expected theoretical curve. The reader is referred to the review by Kapral [30] for more details. 

In this chapter, we discuss the principles of how to calculate fluid flow. As we shall see, hydrodynamics is governed by a partial differential equation, the Navier -Stokes equation. It can be solved analytically only for a few simple cases. A systematic introduction into hydrodynamics is beyond the scope of this book. For an instructive introduction, we recommend Refs [625, 626]. New methods for the calculation of hydrodynamic interactions in dispersions are described in Ref. [627]. As one important example, we derive the hydrodynamic force between a rigid sphere and a plane in an incompressible liquid. Finally, hydrodynamic interactions between fluid boundaries are discussed. 

The situation is more complicated if the sphere approaching a planar surface is deformable, such as a bubble in a liquid or an oil drop in water. We can also think of the inverse situation a solid particle interacting with liquid surface such as a bubble or drop. A particle approaching a liquid-fluid interface will lead to a deformation of the interface. Then, there are three possibilities The particle is repelled by the interface and remains in the first liquid, it goes into the interface and forms a stable three-phase contact, or it crosses the interface and enters the fluid phase completely. The second fluid can be a gas. An example, is interaction of particles with bubbles in a liquid [685]. This interaction is of fundamental importance for flotation [591]. Another example is bubble interacting in a liquid. The hydrodynamic interaction between fluid interfaces is more complicated than between rigid interfaces because we have to take a deformation into account. 

Multiparticle collision dynamics provides an ideal way to simulate the motion of small self-propelled objects since the interaction between the solvent and the motor can be specified and hydrodynamic effects are taken into account automatically. It has been used to investigate the self-propelled motion of swimmers composed of linked beads that undergo non-time-reversible cyclic motion [116] and chemically powered nanodimers [117]. The chemically powered nanodimers can serve as models for the motions of the bimetallic nanodimers discussed earlier. The nanodimers are made from two spheres separated by a fixed distance R dissolved in a solvent of A and B molecules. One dimer sphere (C) catalyzes the irreversible reaction A + C B I C, while nonreactive interactions occur with the noncatalytic sphere (N). The nanodimer and reactive events are shown in Fig. 22. The A and B species interact with the nanodimer spheres through repulsive Lennard-Jones (LJ) potentials in Eq. (76). The MPC simulations assume that the potentials satisfy Vca = Vcb = Vna, with c.,t and Vnb with 3- The A molecules react to form B molecules when they approach the catalytic sphere within the interaction distance r < rc. The B molecules produced in the reaction interact differently with the catalytic and noncatalytic spheres. 

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

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